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To compare the frequencies above and below the median class, let's first identify the key elements required for this comparison:
1. Total Frequency: This is the sum of all frequencies.
2. Median Frequency: This is half of the total frequency.
3. Median Class: This is the class interval in which the median frequency falls.
4. Frequency Below Median Class: This is the count of frequency entries below the median class.
5. Frequency Above Median Class: This is the count of frequency entries above the median class.
Given the table:
| Marks | [tex]$0-10$[/tex] | [tex]$0-20$[/tex] | [tex]$0-30$[/tex] | [tex]$0-40$[/tex] | [tex]$0-50$[/tex] | [tex]$0-60$[/tex] |
|-------|--------|--------|--------|--------|--------|--------|
| Frequency | 10 | 18 | 24 | 36 | 41 | 50 |
Let's proceed with the step-by-step solution:
### Step 1: Determine the Total Frequency
The total frequency is the cumulative frequency at the last class interval (`0-60`):
[tex]\[ \text{Total Frequency} = 50 \][/tex]
### Step 2: Determine the Median Frequency
The median frequency is half of the total frequency:
[tex]\[ \text{Median Frequency} = \frac{50}{2} = 25.0 \][/tex]
### Step 3: Determine the Median Class
The median class is the class interval where the cumulative frequency first exceeds the median frequency (25.0). Examining the cumulative frequencies:
- 10 (for [tex]$0-10$[/tex])
- 18 (for [tex]$0-20$[/tex])
- 24 (for [tex]$0-30$[/tex])
- 36 (for [tex]$0-40$[/tex]) — This is where the cumulative frequency first exceeds 25.0.
Thus, the median class is:
[tex]\[ \text{Median Class} = 0-40 \][/tex]
### Step 4: Count Frequencies Below the Median Class
The number of class intervals below the median class (`0-40`) is 3 (`[tex]$0-10$[/tex]`, `[tex]$0-20$[/tex]`, `[tex]$0-30$[/tex]`).
### Step 5: Count Frequencies Above the Median Class
The number of class intervals above the median class (`0-40`) is 2 (`[tex]$0-50$[/tex]`, `[tex]$0-60$[/tex]`).
### Conclusion
Let's summarize the findings:
- Total Frequency: 50
- Median Frequency: 25.0
- Median Class Index: 3 (corresponding to [tex]$0-40$[/tex])
- Median Class: [tex]$0-40$[/tex]
- Frequency Below Median Class: 3 intervals
- Frequency Above Median Class: 2 intervals
Comparing the frequencies:
- There are 3 class intervals below the median class.
- There are 2 class intervals above the median class.
Thus, there are more frequency class intervals below the median class than above the median class.
1. Total Frequency: This is the sum of all frequencies.
2. Median Frequency: This is half of the total frequency.
3. Median Class: This is the class interval in which the median frequency falls.
4. Frequency Below Median Class: This is the count of frequency entries below the median class.
5. Frequency Above Median Class: This is the count of frequency entries above the median class.
Given the table:
| Marks | [tex]$0-10$[/tex] | [tex]$0-20$[/tex] | [tex]$0-30$[/tex] | [tex]$0-40$[/tex] | [tex]$0-50$[/tex] | [tex]$0-60$[/tex] |
|-------|--------|--------|--------|--------|--------|--------|
| Frequency | 10 | 18 | 24 | 36 | 41 | 50 |
Let's proceed with the step-by-step solution:
### Step 1: Determine the Total Frequency
The total frequency is the cumulative frequency at the last class interval (`0-60`):
[tex]\[ \text{Total Frequency} = 50 \][/tex]
### Step 2: Determine the Median Frequency
The median frequency is half of the total frequency:
[tex]\[ \text{Median Frequency} = \frac{50}{2} = 25.0 \][/tex]
### Step 3: Determine the Median Class
The median class is the class interval where the cumulative frequency first exceeds the median frequency (25.0). Examining the cumulative frequencies:
- 10 (for [tex]$0-10$[/tex])
- 18 (for [tex]$0-20$[/tex])
- 24 (for [tex]$0-30$[/tex])
- 36 (for [tex]$0-40$[/tex]) — This is where the cumulative frequency first exceeds 25.0.
Thus, the median class is:
[tex]\[ \text{Median Class} = 0-40 \][/tex]
### Step 4: Count Frequencies Below the Median Class
The number of class intervals below the median class (`0-40`) is 3 (`[tex]$0-10$[/tex]`, `[tex]$0-20$[/tex]`, `[tex]$0-30$[/tex]`).
### Step 5: Count Frequencies Above the Median Class
The number of class intervals above the median class (`0-40`) is 2 (`[tex]$0-50$[/tex]`, `[tex]$0-60$[/tex]`).
### Conclusion
Let's summarize the findings:
- Total Frequency: 50
- Median Frequency: 25.0
- Median Class Index: 3 (corresponding to [tex]$0-40$[/tex])
- Median Class: [tex]$0-40$[/tex]
- Frequency Below Median Class: 3 intervals
- Frequency Above Median Class: 2 intervals
Comparing the frequencies:
- There are 3 class intervals below the median class.
- There are 2 class intervals above the median class.
Thus, there are more frequency class intervals below the median class than above the median class.
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